One-Day Workshop onApproximation Algorithms and Geometric Convexity

Invited speakers

About hundred years ago, answering a question of Riemann, Steinitz proved the following result. Let B be the unit ball of the Euclidean norm in Rd and assume that V⊆B is finite and the sum of the elements in V is zero. Then there is an ordering of the vectors in V such that all partial sums along this ordering have norm smaller than 2d. I am going to talk about extensions, generalizations of this remarkable theorem and its applications in various fields of mathematics and operations research.

Throughout this abstract, let K be a fat convex body of unit diameter
in d-dimensional space for fixed d and ε > 0 be an arbitrarily small
parameter that controls the approximation error. We recently showed
that a packing of disjoint Macbeath regions of K, all having width ε,
consists of at most O(1/ε^{(d-1)/2}) such regions. The main objective
of this talk is to present three applications of this result to
geometric approximation. We will start with the definition of Macbeath
regions, their properties, and the idea of the proof of the number of
disjoint Macbeath regions of width ε. We then proceed to the
applications.

The first application is in discrete geometry. Approximating convex
bodies succinctly by polytopes is a fundamental problem in the field.
We are given K and ε and the objective is to determine a polytope of
minimum combinatorial complexity whose Hausdorff distance from K is at
most ε. By combinatorial complexity we mean the total number of faces
of all dimensions of the polytope. A well-known result by Dudley
implies that O(1/ε^{(d-1)/2}) facets suffice, and a dual result by
Bronshteyn and Ivanov similarly bounds the number of vertices, but
neither result bounds the total combinatorial complexity. We show that
there exists an approximating polytope whose total combinatorial
complexity is Õ(1/ε^{(d-1)/2}), where Õ conceals a polylogarithmic
factor in 1/ε.
The second application consists of a data structure problem. In the
polytope membership problem we are given K and the objective is to
preprocess K into a data structure so that, given any query point q,
it is possible to determine efficiently whether q is inside K. We
consider this problem in an approximate setting. Given an
approximation parameter ε, the query can be answered either way if the
distance from q to K's boundary is at most ε. We present an optimal
data structure that achieves logarithmic query time with storage of
only O(1/ε^{(d-1)/2}). Our data structure is based on a hierarchy of
Macbeath regions, where each level of the hierarchy is a packing of
Macbeath regions of width δ, and the levels are defined for δ
decreasing exponentially from 1 to ε. This data structure has major
implications to the complexity of approximate nearest neighbor
searching.
The third and last application is an algorithmic problem. Given a set
S of n points and a parameter ε, an ε-kernel is a subset of S whose
directional width is at least (1-ε) times the directional width of S,
for all possible directions. Kernels provide an approximation to the
convex hull, and therefore are on the basis of several geometric
algorithms. We show how the previous hierarchy of Macbeath regions can
be used to construct an ε-kernel in near-optimal time of O(n log(1/ε)
+ 1/ε^{(d-1)/2}). As a consequence, we obtain major improvements to
the complexity of other fundamental problems, such as approximate
diameter, approximate bichromatic closest pair, and approximate
Euclidean minimum bottleneck tree, as well as near-optimal
preprocessing times to multiple data structures.

The problem consists of a geometric system, like a system of lines, hyperplanes or spheres, together with n input points in Rd, and a positive integer k. The objective is to find a subset of at least k input points in general position with respect to the specified system. For example, a set of points is in general position with respect to a system of hyperplanes in Rd if no d+1 points lie on the same hyperplane.

We present a new algebraic framework for solving the Geometric Subset General Position problem and its variants. Our framework provides a unified approach to all the known variants that have been studied in the literature, generalizing them from the planar to the d-dimensional case, as well as enabling the treatment of a large class of new problems, such as Subset General Position with respect to Algebraic Curves, d-Polynomial Subset General Position, Subset Delaunay Triangulation, Circle Subset General Position etc. along with their analogues in projective, spherical and hyperbolic geometry.